Synthetic Apertures for Long-Range, Sub-Diffraction Limited Visible Imaging Using Fourier Ptychography

ABSTRACT

A method for imaging objects includes illuminating an object with a light source of an imaging device, and receiving an illumination field reflected by the object. An aperture field that intercepts a pupil of the imaging device is an optical propagation of the illumination field at an aperture plane. The method includes receiving a portion of the aperture field onto a camera sensor, and receiving a sensor field of optical intensity. The method also includes iteratively centering the camera focus along the Fourier plane at different locations to produce a series of sensor fields and stitching together the sensor fields in the Fourier domain to generate an image. The method also includes determining a plurality of phase information for each sensor field in the series of sensor fields, applying the plurality of phase information to the image, receiving a plurality of illumination fields reflected by the object, and denoising the intensity of plurality of illumination fields using Fourier ptychography.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application incorporates by reference and claims the benefit ofpriority to U.S. Provisional Patent Application No. 62/532,637 filedJul. 14, 2017.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under CCF1117939,CCF1527501, CNS1338099, IIS1116718, and IIS1453192 awarded by theNational Science Foundation; and HR0011-16-C-0028 awarded by the DefenseAdvanced Research Projects Agency (DARPA); N00014-14-1-0741 (SubcontractGG010550 from Columbia University) and N00014-15-1-2735 awarded by theOffice of Naval Research. The government has certain rights in theinvention.

BACKGROUND OF THE INVENTION

The present subject matter relates generally to synthetic apertures forvisible imaging. More specifically, the present invention relates tosynthetic apertures for visible imaging as a promising approach toachieve sub-diffraction resolution in long distance imaging.

Imaging objects from large standoff distances is a requirement in manycomputer vision and imaging applications such as surveillance and remotesensing. In these scenarios, the imaging device is sufficiently far awayfrom the object that imaging resolution is fundamentally limited not bymagnification, but rather by the diffraction of light at the limitingaperture of the imaging system: using a lens with a larger aperture willlead to increased spatial resolution. Physically increasing the apertureof the lens, by building a larger lens, results in expensive, heavy, andbulky optics and mechanics. A number of techniques have been proposed toimprove spatial resolution for various imaging systems, includingrefractive telescopes (1-6), holography (7-11) and incoherentsuper-resolution (12-16).

The resolution of an imaging system is proportional to both the lensaperture size and the wavelength of the electromagnetic spectrum used.In long wavelength regimes (such as radar), the direct coupling betweenimage resolution and aperture size can be mitigated using syntheticaperture radar (SAR) techniques. SAR improves radio imaging resolutionby capturing multiple measurements of a static object using a mobilerecording platform, such as an airplane or satellite as shown in thediagram of FIG. 1A. For SAR, the resolution is determined by thesynthetic aperture size, which can be many orders of magnitude largerthan the physical aperture size. Stitching together multiple radarreturns is possible because the full complex field (amplitude and phase)is directly measured by the antenna with picosecond timing resolution.

As noted above, stitching together multiple radar returns from a SARtechnique is possible because the amplitude and phase is measured withpicosecond timing resolution. To make a comparable measurement usingvisible light, a detector would have to continuously record informationwith a time resolution greater than one femtosecond, a requirement wellbeyond the capabilities of modern devices. As such, current camerasensors record only the intensity of the incoming optical field and allphase information is lost.

Fourier ptychography (FP) has emerged as a powerful tool to improvespatial resolution in microscopy. In FP, multiple low-resolution imagesunder different illumination angles are captured and stitched together(17-27). Redundancy between measurements permits computational recoveryof the missing phase information (18,25, 28-31). Fourier ptychographycreates a synthetic aperture by sampling a diverse set of regions inFourier space. Unlike holography, FP does not require the use of areference beam to encode phase information. The phase of the complexfield is recovered computationally in post-processing. FP has found muchof its success in microscopy. Early efforts by Kirkland et al. (59, 60)demonstrated that multiple images recorded with different incident beamtilts could be used to effectively double image resolution. Zheng et al.(17) provided a complete framework for FP microscopy and demonstratedwide-field, high-resolution imaging. Subsequent research has improvedthe quality of FP reconstructions by characterizing the pupil function(18), digitally removing optical aberrations (19), and refocusing therecovered image postcapture (20). FP microscopy (where the illuminationdirection is varied) inherently assumes that the sample may be modeledas a thin object. Extensions for thick biological samples (21-23) havebeen proposed at the expense of increased computational complexity.

At the heart of FP is the requirement to recover the phase of the lightfield at the aperture plane of the lens, which subsequently providesknowledge of the field at the object plane. Phase retrieval is also animportant step in standard and many of the techniques used in FP areborrowed from these earlier efforts.

In general, closed form solutions for recovering phase informationrequire prohibitively large datasets to be practical (61-63). Iterativesolutions are thus preferred for ptychographic reconstruction. Many FPreconstruction algorithms are based on the iterative update schemesfirst proposed by Gerchberg and Saxton (28) and Fienup (29). Maiden andRodenburg (30) introduced the ePIE technique to jointly estimate thefield at the detector and the probe used for illumination. Ou et al.(18) adapted ePIE for use in FP whereby the pupil function is jointlyestimated with the field at the aperture plane. Experimental robustnessof various phase retrieval algorithms were characterized by Yeh et al.(31) who conclude that minimizing the error in amplitude and usingsecond-order gradient descent methods provide the best results. Thephase retrieval algorithm used by Tian et al. (25), which incorporatesthe pupil update step of (18) and uses the second-order Newton's methodas the numerical solver, serves as the base of our proposed algorithm.Although the objective function of the reconstruction framework in (25)minimizes intensities and not amplitudes, our experiments have resultedin good reconstruction quality.

Adapting the technique to long-distance imaging requires two importantmodifications of previous FP microscopy implementations. First, theseparation distance between object and camera increases by orders ofmagnitude. Second, a reflection imaging geometry must be used so thatillumination source and camera are placed on the same side of theobject. Dong et al. (20) and Holloway et al. (32) succeeded in the firsttask, scaling up the working distance to 0.7 and 1.5 meters,respectively. Reflective FP microscopy setups have been proposed tofulfill the second task (33-35). However, these systems either requiresmall working distances (34, 35), or exhibit limited reconstructionperformance (33).

In FIGS. 1B and 1C, a comparison with existing FP implementations isshown. Previous works have relied on smooth objects and are looselyrepresented by the transmissive dataset adapted from (32) shown in FIG.1B. An example dataset of a diffuse object collected in a reflectionmode geometry is shown in FIG. 1C. The immediate difference between thetwo datasets is the random phase associated with diffuse objectseffectively spreads out information across the entire Fourier domain.The difference in Fourier patterns is evident in the captured imagestaken from the same locations in both modalities. As a consequence ofthe random phase, the spatial information is obfuscated by the resultantspeckle.

Tippie et al. (11) proposes a synthetic aperture holographic setup inwhich the authors experimentally demonstrated synthetic apertureoff-axis holographic capture of diffuse objects at a large stand-offdistance. Our approach can be interpreted as a reference-free extensionof synthetic aperture holography in which computational reconstructionalgorithms are used in place of interferometric capture, resulting inmore stable implementations and widening the variety of applicationscenarios that could benefit from the approach. Beck et al. (36)proposes an optical synthetic aperture approach that extends SARtechniques into optical wavelengths in the near-infrared regime of theelectromagnetic spectrum. To record phase measurements, the object israster scanned by moving an aperture. The return signal is thendemodulated using a reference signal to reduce the frequency toapproximately 100 kHz, which can be sampled with commercially availableADCs.

BRIEF SUMMARY OF THE INVENTION

The present application discloses a method for imaging objects usingmacroscopic Fourier ptychography (FP) as a practical means of creating asynthetic aperture for visible imaging to achieve sub-diffractionlimited resolution. Also disclosed is a novel image space denoisingregularization to reduce the effects of speckle and improve perceptualquality of the recovered high-resolution image, resulting in a 6×improvement in image resolution.

In one embodiment, a method for imaging objects includes the steps ofproviding an imaging device including a camera sensor, a camera lens,and a pupil; illuminating an object with the light source; receiving anillumination field reflected by the object, wherein an aperture fieldthat intercepts the pupil of the imaging system is an opticalpropagation of the illumination field at an aperture plane; receiving aportion of the aperture field onto a camera sensor; receiving a sensorfield of optical intensity on the image sensor, wherein the sensor fieldis a Fourier transform of the aperture field immediately after thecamera lens; iteratively centering the camera focus along the Fourierplane at different locations to produce a series of sensor fields;stitching together the sensor fields in the Fourier domain to generatean image; determining a plurality of phase information for each sensorfield in the series of sensor fields; applying the plurality of phaseinformation to the image; receiving a plurality of illumination fieldsreflected by the object; and denoising the intensity of plurality ofillumination fields using Fourier ptychography.

In a further embodiment the optical propagation is one of a Fouriertransform, a Fresnel transform, an angular spectrum propagation, and aHuygens convolution. In a yet further embodiment, the object includes anoptically rough surface such that the illumination fields of theplurality of illumination fields are out of phase, creating speckle.

In another embodiment, the method further includes the step of utilizingFourier ptychography to reduce speckle by increasing the aperturediameter and/or the step of inserting a rotating diffuser in an opticalpath before the object to reduce diffraction blur. Further, the methodmay include the step of generating a grid of images to yield a syntheticaperture. In some embodiments, a synthetic aperture of at least 14.5 mmis generated.

In a further embodiment, the method includes the steps of recordingimages with three different shutter times and joining the recordedimages together to yield a high-dynamic range image. In a still furtherembodiment, the step of determining a plurality of phase information foreach sensor field comprises iteratively estimating the image intensityin the Fourier domain.

Additional objects, advantages and novel features of the examples willbe set forth in part in the description which follows, and in part willbecome apparent to those skilled in the art upon examination of thefollowing description and the accompanying drawings or may be learned byproduction or operation of the examples. The objects and advantages ofthe concepts may be realized and attained by means of the methodologies,instrumentalities and combinations particularly pointed out in theappended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawing figures depict one or more implementations in accord withthe present concepts, by way of example only, not by way of limitations.In the figures, like reference numerals refer to the same or similarelements.

FIG. 1A is a diagram illustrating a prior art mobile radar imagingsystem utilizing a synthetic aperture radar (SAR).

FIGS. 1B and 1C illustrate a simulation of recovering smooth resolutiontarget in a transmissive geometry and a simulation of recovering a roughresolution target in reflective geometry, respectively.

FIGS. 2A and 2B are diagrams illustrating a synthetic aperture visibleimaging (SAVI) technique and a SAVI technique utilizing Fourierptychography.

FIG. 3 is a flow chart of a recovery algorithm to suppress speckleduring the reconstruction step of the SAVI technique shown in FIG. 2B.

FIG. 4 is images illustrating the application of the recovery algorithmof FIG. 3.

FIG. 5 illustrates a diagram of a system for imaging objects.

FIGS. 6A and 6B illustrate results of utilizing the system of FIG. 5.

FIG. 7 illustrates examples of a captured image at each stage of thesystem of FIG. 5.

FIG. 8 illustrates detailed regions at each stage illustrated in FIG. 7.

FIGS. 9A and 9B are graphs illustrating contrast as a function offeature size and speckle size relative to aperture diameter,respectively, of the images of FIGS. 7 and 8.

FIG. 10 illustrates simulated reconstructions for varyingbackground/foreground contrast ratios.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 2A is a diagram illustrating a synthetic aperture for visibleimaging (SAVI) technique 100. In a first step 102, a coherent source104, e.g. a laser, illuminates a distant target 106 and the reflectedsignal is captured by a camera system 108 in intensity images 110. Thecamera system 108 is translated to capture all of the light that wouldenter the desired synthetic aperture in a second step 112. Unlike SAR,phase information cannot be recorded for visible light. High-resolutionimage reconstruction therefore necessitates post-capture computationalphase recovery in a third step 114.

FIG. 2B illustrates a further diagram of SAVI 200 in three steps. In thefirst step 202, a diffuse object 204 is illuminated with a coherentsource 206 and overlapping images 208 are captured. Each image islow-resolution and suffers from speckle. In step 210, missing phaseinformation is recovered computationally, which requires redundancybetween captured intensity images. In step 212, a high-resolution imageis reconstructed using the recovered phase. Additional examples ofcaptured data and reconstructions are shown in FIGS. 6A-8.

Section I. Materials and Methods: Reconstructing Diffuse Objects

Fourier ptychography relies on the use of monochromatic illuminationsources to provide coherent illumination. An overview of the forwardmodel of FP is provided here.

A. Image Formation Model

In this embodiment, a source illuminates an object which reflects lightback toward a camera (see FIG. 5). The source emits a field that ismonochromatic with wavelength λ and is spatially coherent across thesurface of the object of interest. The illumination field interacts withthe object and a portion of the field will be reflected back toward theimaging system. The field emanating from the object is a constant 2Dcomplex wave, ψ(x, y).

ψ(x, y) propagates over a sufficiently large distance z toward theimaging system to satisfy the far-field Fraunhofer approximation. Thefield at the aperture plane of the camera is related to the field at theobject through a Fourier transform (37):

$\begin{matrix}{{{\Psi \left( {u,v} \right)} = {\frac{e^{jkz}{e^{{jk}\text{/}2z}\left( {u^{2} + {v^{2}\_}} \right.}}{j\; \lambda \; z}F_{\frac{1}{\lambda \; z}}\left\{ {\psi \left( {x,y} \right)} \right\}}},} & (1)\end{matrix}$

where k=2πλ is the wavenumber and is the two-dimensional Fouriertransform scaled by 1/λz. For simplicity, the multiplicative phasefactors and the coordinate scaling have been excluded from the analysis,though these can be accounted for after phase retrieval if desired. Tofurther reduce clutter, spatial coordinates (x, y) will be representedby the vector x and frequency coordinates (u, v) will be represented asu.

The field that intercepts the pupil of the imaging system, Ψ(u), iseffectively the Fourier transform of the field at the object plane. Dueto the finite diameter of the lens, only a portion of the Fouriertransform is imaged onto the camera sensor. Let the transmittance of thepupil, be given by P(u). For an ideal circular lens, P(u) would bedefined as:

$\begin{matrix}{{P^{m}(u)} = \left\{ {\begin{matrix}{{{1\mspace{14mu} {if}\mspace{14mu} {u}} < \frac{d}{2}},} \\{{0\mspace{14mu} {otherwise}}\mspace{20mu}}\end{matrix},} \right.} & (2)\end{matrix}$

where d is the diameter of the lens.

The camera lens is focused on the image sensor, and therefore alsofulfills the Fraunhofer approximation (37), such that the field at thesensor plane (again ignoring phase offsets and scaling) is the Fouriertransform of the field immediately after the lens. Since the imagesensor only records optical intensity, the final image is given as

I(x, c)∝|F{Ψ(u−c)P(u)}|²,  (3)

where c is the center of the pupil. In Eq. (3), the shift of the cameraaperture to capture different regions of the Fourier domain isrepresented by the equivalent shift of the Fourier pattern relative tothe camera. Due to the finite extent of the lens aperture, the recordedimage will not contain all of the frequency content of the propagatedfield Ψ(u). For a lens with diameter d and focal length f, the smallestresolvable feature within one image will be approximately 1.22λƒ/d.

B. Recovering High-Resolution Magnitude and Phase

To facilitate the creation of a synthetic aperture, the camera isre-centered in the Fourier plane at N different locations c_(i), i=1, .. . , N. One consequence of sampling in the Fourier domain is that theimages must be stitched together in the Fourier domain. From Eq. (3),the image sensor only records the intensity of the complex field andcontains no information about the phase. It is therefore necessary tocomputationally recover the phase of the N intensity measurements. Toensure sufficient information is available for post-capture phaseretrieval, care must be taken to provide sufficient overlap betweenadjacent camera positions. In some embodiments, a minimum of about 65%overlap is typically required in order for phase retrieval to convergeto an adequate solution (18, 25, 32).

From Eq. (1), recovering Ψ(u) effectively recovers ψ(x) as the fieldsare related through a Fourier transform. The present system seeks tosolve the following optimization problem:

$\begin{matrix}{{\hat{\Psi}(u)},{{\hat{P}(u)} = {\min\limits_{{\Psi {(u)}},{P{(u)}}}{\sum\limits_{i}{{{I\left( {x,c_{i}} \right)} - {{F\left\{ {\Phi \left( {u,c_{i}} \right)} \right\}}}^{2}}}}}},} & (4)\end{matrix}$

where Φ(u, c_(i)) is the optical field immediately following theaperture at the ith position, Φ(u, c_(i))=Ψ(u−ci)P(u).

Defining the data fidelity term of the reconstruction to be the L₂ errorbetween measured and estimated intensities in Eq. (4) results in anon-convex optimization problem. Phase retrieval is typically solvedusing an iterative update scheme similar to those popularized byGerchberg-Saxton (38) and Fienup (39). In the mth iteration, theestimate of Ψ(u) is propagated to the image plane for each cameraposition c_(i), whereupon the measured image intensities are enforced:

$\begin{matrix}{{{\Phi^{m}\left( {u,c_{i}} \right)} = {{\Psi^{m}\left( {u - c_{i}} \right)}{P(u)}}},} & (5) \\{{{\varphi^{m}\left( {x,c_{i}} \right)} = {F^{- 1}\left\{ {\Phi^{m}\left( {u,c_{i}} \right)} \right\}}},} & (6) \\{{{\varphi^{m + 1}\left( {x,c_{i}} \right)} = {\sqrt{I\left( {x,c_{i}} \right)}\frac{\varphi^{m}\left( {x,c_{i}} \right)}{{\varphi^{m}\left( {x,c_{i}} \right)}}}},} & (7) \\{{\Phi^{m + 1}\left( {u,c_{i}} \right)} = {F{\left\{ {\varphi^{m + 1}\left( {x,c_{i}} \right)} \right\}.}}} & (8)\end{matrix}$

Differences between successive estimates of the field Ψ(u, c_(i)) areused to update the estimates of Ψ(u) and P(u) in the Fourier domain.Following the formulation in (25), the estimate of Ψ(u) is given by,

$\begin{matrix}{{\Psi^{m + 1}(u)} = ~{{\Psi^{m}(u)} + {\frac{{P^{m}\left( {u + c_{i}} \right)}}{{{P^{m}(u)}}_{\max}}\frac{\left\lbrack {P^{m}\left( {u + c_{i}} \right)} \right\rbrack^{*}}{{{P\left( {u + c_{i}} \right)}}^{2} + \tau_{1}} \times {\left\lbrack {{\Phi^{m + 1}\left( {u - c_{i}} \right)} - {\Phi^{m}\left( {u - c_{i}} \right)}} \right\rbrack.}}}} & (9)\end{matrix}$

The adaptive update step size, |P^(m)(u+c_(i))|/|P^(m)(u)|_(max), isused in (25) and is based on the work of Rodenburg and Faulkner (40).The contribution of the pupil function is first divided out of thedifference and the remainder is used to update the estimate of Ψ(u). Asimilar update step is used to update the estimate of the pupil functionbut with the roles of Ψ(u) and P(u) reversed,

$\begin{matrix}{{P^{m + 1}(u)} = ~{{P^{m}(u)} + {\frac{{\Psi^{m}\left( {u - c_{i}} \right)}}{{{\Psi^{m}(u)}}_{\max}}\frac{\left\lbrack {\Psi^{m}\left( {u - c_{i}} \right)} \right\rbrack^{*}}{{{\Psi \left( {u - c_{i}} \right)}}^{2} + \tau_{2}} \times {\left\lbrack {{\Phi^{m + 1}(u)} - {\Phi^{m}(u)}} \right\rbrack.}}}} & (10)\end{matrix}$

In the update steps shown in Eq. (9) and Eq. (10), a small value (τ₁ andτ₂ respectively) is added to prevent division by zero. In (18), theupdated pupil function was constrained to lie within the support of theinitial guess, which corresponds to the numerical aperture of the lens.The support is twice as large as the initial guess to accommodatedifferences between the experimental setup and the forward model (suchas the aperture not being a perfect circle).

Initial estimates of Ψ(u) and P(u) must be provided. The initialestimate of the pupil function is defined to be an ideal circularaperture from Eq. (2) with a diameter determined by the aperture. Acommon initialization of Ψ(u) for weakly scattering objects is toupsample any of the recorded images (often an image near the DCcomponent) and take its Fourier transform (17, 18, 25). In oneembodiment, Ψ(u) is the Fourier transform of the average of all Ncaptured intensity images. Averaging independent measurements of thefield suppresses speckle in the initial estimate (1, 3).

C. Optically Rough Objects

Typical applications of FP in microscopy have dealt with objects thathave gradual changes in refractive index, which leads to transferfunctions with relatively smooth phase components. However, the surfacesof most real-world objects are “optically rough” and exhibit randomphase.

When coherent light reflects off of an object, each point along thesurface acts as a secondary source of spherical illumination. Theconstituent components of the reflected optical field will be comprisedof a summation of each of the secondary sources. If the variation inpath length exceeds the wavelength of the incident light, λ˜550 nm, thesecondary sources will be out of phase with one another. Summation ofthe dephased point sources leads to destructive interference whichmanifests as speckle (41, 42).

Suppose that the variation of surface height is at least equal to λ andis uniformly distributed. For any point in the optical field, theprobability of measuring an intensity I (squared amplitude) follows anegative exponential distribution (43):

$\begin{matrix}{{{p(I)} = {{\frac{1}{\mu}e^{- \frac{I}{\mu}}\mspace{14mu} {for}\mspace{14mu} I} \geq 0}},} & (11)\end{matrix}$

where μ is the mean intensity. It should be noted that Eq. (11) holdsfor fully-developed speckle whereby polarized light maintains itspolarization state after reflection. Most real-world objects exhibitsubsurface scattering that destroys the polarization state of theincident light. In such a case the intensity distribution is given as(44):

$\begin{matrix}{{p(I)} = {{4\frac{1}{\mu^{2}}e^{{- 2}\frac{I}{\mu}}\mspace{14mu} {for}\mspace{14mu} I} \geq 0.}} & (12)\end{matrix}$

For the purposes of this paper it is sufficient to say that speckleintensity follows a negative exponential distribution.

In an imaging geometry, the apparent speckle size is compounded bydiffraction blur induced by the aperture of the lens. As such, thespeckle size at the sensor plane is approximately twice that of thesmallest resolvable image features (2.44λƒ/d) (43). Fourier ptychographyis used to reduce diffraction blur by synthetically increasing theaperture diameter. In the presence of speckle, FP also reduces specklesize.

The formation of speckle is compatible with the analysis of theformation model given here and used in other previous work, and in factis a natural consequence of the object having a randomly distributedphase. Previous FP implementations have generally avoided dealing withspeckle by either imaging thin biological samples which naturally havesmooth phase, by using partially coherent light (45), or by acombination of the two. The present application provides a macroscopicFP imaging system that recovers optically rough objects.

D. Suppressing Speckle

Fourier Ptychography reduces speckle size by reducing diffraction blur,however the variation in the remaining speckle is still large. Therecovery algorithm described in Section II-B below leads to thesuppression of speckle during reconstruction in the estimate of Ψ(u).Section II-B discusses the comparison of the recovery algorithm of themethod of the present application with an alternate speckle suppressiontechnique of averaging independent measurements (1, 3).

From Eq. (11) and Eq. (12), speckle intensity follows a negativeexponential distribution, which is consistent with a multiplicativenoise model. It is important to note that speckle is not noise in theconventional sense. The underlying random phase of the object distortsthe intensity field recorded by the sensor. It is desirable to mitigateany distortion that manifests as speckle in order to recover ahigh-resolution intensity image.¹ In this sense, speckle is referred toas “noise.”¹ Recovering a high-resolution estimate of the magnitude andphase may be useful for determining material properties and may beaccomplished by running a second reconstruction that omits the specklesuppression step.

Much of the research related to image denoising techniques, includingthe state-of-the-art BM3D algorithm (46), assumes an additive noisemodel. The intensity of Ψ(x) is denoised during image recovery in orderto recover a high quality intensity image. To overcome themultiplicative distortions caused by speckle, the noise is converted bytaking the natural logarithm of the intensity at iteration m into a moreconvenient additive model:

ξ^(m)(x)=ln(|ψ^(m)(x)|²),  (13)

where the new variable ξ^(m)(x) is modeled as the true (noise free)signal ξ′(x) corrupted by additive noise η(x),

ξ^(m)(x)=ξ′(x)+η(x).  (14)

Extracting an updated estimate, ξ^(m+1)(x), that better approximatesξ′(x) can be achieved using any of the standard image denoisingtechniques. The following straightforward image noise suppression methodmay be used in the present application: wavelet decomposition followedby soft-thresholding wavelet components whose magnitudes exceed ρ (47).Denoising is accomplished by decomposing ξ using an orthogonal waveletbasis W and denoising operator D(α, ρ):

ξ^(m+1)(r)=W[D(W ^(T)ξ^(m)(x), ρ)],  (15)

D(α, ρ)=sgn(α)max(0,|α|−ρ).  (16)

Symlet wavelets of order eight are used as the basis for waveletdecomposition and decompose to the second level. The value of ρ may beset a priori or may be updated automatically, e.g. using Stein'sunbiased risk estimator (SURE) (48). In the embodiment described below,the latter method is adopted. Finally, the amplitude of ψ^(m+1)(x) isupdated in a manner similar to Eq. (8),

$\begin{matrix}{{{\psi^{m + 1}(x)} = {\sqrt{e^{\xi^{m + 1}{(x)}}}\frac{\psi^{m}(x)}{{\psi^{m}(x)}}}},} & (17)\end{matrix}$

which is then transformed back into the Fourier domain as Ψ^(m+1)(u) andthe recovery algorithm proceeds as previously described. In oneembodiment of the speckle suppression, large coefficients in the waveletdomain are removed, which is similar to imposing a histogram constraintcommonly used in phase retrieval (49) in the wavelet domain.

Denoising is applied every s iterations of the recovery algorithm.Spacing out denoising allows adequate time for information transfer andreinforcement between the spatial and Fourier domains during phaseretrieval. The recovery algorithm effectively has an outer loop toperform the denoising, and an inner loop for phase retrieval.

A visual representation of the complete recovery algorithm is shown inFIG. 3. A high resolution estimate of Ψ(u) is recovered by iterativelyenforcing intensity measurement constraints in the spatial domain andupdating estimates of the spectrum in the Fourier domain. Imagedenoising is is applied every s iterations to suppress the influence ofspeckle noise in the final reconstruction. The branch on the left sideof FIG. 3 is a traditional FP recovery algorithm used in (18, 25). Thebrand on the right side of FIG. 3 is image denoising to reduce specklenoise in the estimate of Ψ(u).

FIG. 4 demonstrates the benefit of incorporating the denoising step ofFIG. 3 into the recovery algorithm in the presence of subjectivespeckle. A simulated high resolution complex object is imaged by adiffraction limited system. The amplitude is a resolution chart shown in(A), and the phase is randomly distributed in the range [−π, π]. Directobservation of the object results in an image degraded by diffractionblur and speckle, one such example is shown in (B). Using FP without thedenoising regularization results in artifacts in the reconstruction asshown in (C). Incorporating the denoising step during reconstruction, asin (D), reduces some of the artifacts present in (C) and improvesoverall contrast by reducing the intensity variation in the speckleregions. The brightness of the outsets in (B)-(D) has been increased tohighlight the presence of reconstruction artifacts.

To compare the performance between (C) and (D), the variance of thewhite owl, σ_(w) ², and the grey text and camera, σ_(g) ², are computed.Without denoising, the variance of the white and grey regions is 44.6and 3.2 pixels respectively (with a maximum intensity of 255). Using theproposed denoising method reduces the variance by 55% to 20.4 and 1.5pixels respectively in the white and grey regions.

Section II Results: Fourier Ptychography in Reflection Geometry

To validate the proposed method of creating synthetic apertures forvisible imaging, a table top device is constructed to capture images ofobjects under coherent illumination. Image recovery is performed inMATLAB.

A. Experimental Setup

A simplified rendering of the experimental setup for reflection geometryFourier ptychography is shown in FIG. 5. The system 300 includes a laserdiode 302 (Edmund Optics #64-825) operating at 532 nm is passed througha spatial filter 304 and a 2-inch diameter focusing lens 306 toilluminate an object 308 positioned about 1 meter away. For clarity, thespatial filter 304 has been represented as a pinhole and the camera bodyhas been removed in FIG. 5. Additional optical elements such as NDfilters and polarizers have also been omitted from the diagram.

Light scattered off of the object 308 is collected by a camera 310positioned near the focusing lens 306. In order to satisfy theFraunhofer diffraction approximation for a short optical path, a lens306 is used to focus the coherent illumination at the aperture plane ofthe camera lens. Multiple images are acquired to create a largesynthetic aperture by translating the camera and lens using atranslation stage. While the model in Section II assumes free spacepropagation, the analysis holds for converging spherical waves. Imagingover larger distances would not require a lens to satisfy the far-fieldconditions, and the focusing lens would be repurposed as a collimatinglens instead. Low-resolution images are captured by moving the camera(both the lens and sensor) on a XY translation stage to sample a largerregion of Fourier space.

In the experiments described herein, the following parameters andcomponents are used. The camera used is a Point Grey machine visioncamera (BFLY-PGE-50A2M-CS) equipped with an Aptina MT9P031 CMOS imagesensor with a pixel pitch of 2.2 μm. In front of the sensor is a 75 mmfocal length lens and an aperture diameter of 2.5 mm (ƒ/30), whichproduces a diffraction spot size of ˜39 μm on the sensor. For the USAFtarget and fingerprint datasets, adjacent positions of the camera are0.7 mm apart to ensure sufficient overlap between samples in the Fourierdomain. A grid of 19×19 images is captured to produce a syntheticaperture of 15.1 mm, six times larger than the lens' aperture. Theparameters used to capture the reverse side of a US $2 bill in FIG. 6Bare slightly modified. A 21×21 grid of images is collected with adjacentpositions separated by 0.6 mm, yielding a synthetic aperture of 14.5 mm.

Exposure bracketing and image averaging are employed to increase thedynamic range and reduce read noise respectively. At each position, thecamera records images with 3 different shutter times (doubling theexposure between each). For each exposure time, between 5-10 images areaveraged to reduce read noise. The exposures are then joined together toyield a high-dynamic range image. An image sensor with a larger ADC andlarger pixels could be substituted to decrease acquisition time insteadof employing averaging and exposure bracketing.

Accurate alignment of the low-resolution images is crucial to accuratelyreconstruct a high-resolution image. Checkerboard fiducial markers areplaced at the periphery of the object, outside of the region illuminatedby coherent light, to allow for the ease of alignment post-capture usingstandard tools (50). If fiducial markers are not present, diffuse imagescan be aligned by correlating speckle patterns in adjacent images (11).In long distance imaging, it is likely that only a portion of the scenewill be illuminated and key features in the rest of the scene may beused for image alignment, which matches the operating parameters of ourexperimental setup.

B. SAVI for Diffuse Objects

FIGS. 6A and 6B provide examples of results using SAVI to improvespatial resolution in long-range imaging for everyday diffuse objects.Images captured with a small aperture are unrecognizable due todiffraction blur and speckle. Following reconstruction, fine details arevisible. In FIGS. 6A and 6B, two common objects are illuminated withcoherent light: in FIG. 6A a fingerprint deposited on glass is presentedand FIG. 6B shows the reverse side of a US $2 bill. The bar represents 1mm in FIG. 6A and 2 mm in FIG. 6B. A single captured image for eitherobject is unrecognizable due to the extreme degradation from speckle anddiffraction blur. After reconstruction using Fourier ptychography, whichamounts to a six-fold improvement in resolution, both images arerecognizable and fine details can be clearly observed.

The ability to resolve biometric markers from large stand-off distanceshas many applications in surveillance and authentication. Currentmethods of data collection are often intrusive, e.g. using a fingerprintscanner. FIG. 6A illustrates how a fingerprint can be recovered from apiece of glass by means of SAVI. The fingerprint is coated with powderto provide relief against the transparent glass. Fine features necessaryto identify the print are recovered as illustrated in the zoomed-indetail regions. Due to the limitations in the current prototype, onlystationary objects can be recovered; however, a real-time acquisitionsystem would enable contactless on-the-fly authentication of userssimply holding a finger aloft.

Capturing legible text and images is also of interest in surveillancescenarios, such as verifying payment amounts at retail outlets. However,recovering the high-resolution image for the $2 bill (FIG. 6B)introduced an unanticipated problem during reconstruction. Unlike thefingerprint, the intensities on the cloth are not binary which resultsin reduced contrast between the foreground and background. To combat theadded difficulty of recovering such a varied object, the overlap betweenadjacent measurements was increased to 76% and a 21×21 grid was sampled.An analysis of recovering scenes with low contrast is discussed inSection IV below.

During phase retrieval, a high-resolution estimate of the phaseemanating from the scene is recovered. As expected for a diffuselyscattering object, the phase exhibits a random profile in the range [−π,π].

USAF resolution target: In order to quantitatively investigate theperformance of SAVI, a negative USAF chrome-on-glass target with flatwhite paint applied to the chrome surface of the target is imaged. Thetarget is imaged through the back of the glass plate to retain thehigh-resolution features characteristic of the resolution chart. Anexample of a captured image is shown in the first row of FIG. 7.

The specular reflections off of the glass and chrome surfaces of thetarget are orders of magnitude stronger than the diffuse light scatteredby the painted surface. To mitigate the effects of specular reflection,the angle between the illumination, object, and camera was adjusted sothat the direct reflection does not enter the synthetic aperture of thecamera system. Additionally, crossed polarizers are used to attenuatethe contributions of diffracted direct illumination (at the boundariesof the bars).

Directly imaging the target results in a characteristic speckle patternas shown in the first row of FIG. 7. For the given imaging geometry,resolution is limited to 1.26 line pairs per millimeter or a bar widthof 397 μm. A common method to reduce speckle noise, and increaseresolution, is to average multiple short exposure images of a varyingspeckle field to extract spatial frequencies beyond the capability ofany single measurement (1, 3). To induce a change in speckle pattern thetarget is vibrated and a sequence of 361 short-exposure images areacquired, equal to the number of images used to create the syntheticaperture. The exposure duration is set to be ⅕ of the middle exposuretime used during FP data collection. The average of the short-exposureimages is shown in the second row of FIG. 7. As expected, the resolutionof the averaged image surpasses that of the captured image and 280 μmfeatures are now resolvable.

Speckle can also be suppressed by inserting a rotating diffuser in theoptical path before the object (51) to destroy the temporal coherence ofthe illumination source. As shown in the third row of FIG. 7, imagingwith a rotating diffuser improves resolution so that features as smallas 157 μm are resolvable. A consequence of using a rotating diffuser,light intensity falls off as 1/d² which greatly impacts the intensitymeasured by the camera and limits the effective working distance of thesystem.

While individual images exhibit significant blur and diffraction, whichcan be partially mitigated by averaging, the resulting image from FP hasa 6× improvement in resolution with resolvable features as small as 70μm (fourth row of FIG. 7). The increased resolution is also accompaniedby a corresponding 6× decrease in speckle size, however the speckle isstill present in the final reconstruction. By introducing the denoisingprior from Eq. (17) into the reconstruction algorithm, the effect of thespeckle can be further mitigated to improve contrast while retaining thesame resolution improvements as traditional FP algorithms. The fullreconstruction with regularization is shown in the fifth row of FIG. 7.

Zoomed in details of the five imaging scenarios of FIG. 7 are shown inFIG. 8. Images above dashed line are not resolvable. Notice that theresolution and contrast quickly deteriorates for individual capturedimages and the averaged image while using the rotating diffuserincreases resolution marginally better. The images obtained using FPexhibit higher resolution than directly measuring the scene;furthermore, the addition of the denoiser reduces variation within thespeckle regions to improve contrast.

The USAF resolution chart can be used to quantify the resolving power ofan optical system. Many metrics to compute contrast rely on the ratio ofthe maximum value to the minimum value along a cross sectionperpendicular to the bar orientation. This is a serviceable definitionin most cases; however, when speckle is present the random distributionof intensities can skew contrast measurements. To account for thevariable intensity due to speckle, bar positions known a priori and theaverage intensities of the white (w) and black bars (b) are used tocompute contrast. The contrast C is further scaled by the averageintensity in the bright and dark regions to account for specklemigration during reconstruction,

$\begin{matrix}{C = {\frac{\overset{\_}{w} - \overset{\_}{b}}{\overset{\_}{w} + \overset{\_}{b}}*\overset{\_}{w}*{\left( {1 - \overset{\_}{b}} \right).}}} & (18)\end{matrix}$

Bar locations are manually located using a high-resolution image of theUSAF target and are scaled to the correct size for each test image. Thethreshold for resolvability must be adjusted to compensate to theadditional scaling in Eq. (18). In some embodiments, a contrast value of0.05 is the resolution limit.

Contrast plots for the USAF target are presented in FIG. 9A. Contrastfor the observation image and the averaged short exposure image rapidlydeteriorate in agreement with observations made in FIGS. 7 and 8.Reconstruction of the synthetic aperture image using the full complementof measurements significantly improves resolution. Including the imagespace denoising during reconstruction increases the contrast in the barswhich aids in discrimination of fine spatial features. Resolution in thecaptured image (1) deteriorates rapidly. Mimicking an incoherent sourcevia averaging (2) or rotating diffuser (3) increases spatial resolution.SAVI without (4) and with (5) a regularizer drastically increasesresolution. Use of the regularizer improves image contrast.

Resolution gains and speckle reduction: As the goal of the proposedmethod is to emulate a larger synthetic aperture, and diffraction blurand speckle size are inversely proportional to the size of the imagingaperture, the improvement in resolution should increase linearly withthe increase in aperture diameter. To illustrate this effect, the USAFtarget was reconstructed with varying synthetic aperture sizes by onlyusing subsets of the full 19×19 dataset. In this manner, the size of thesynthetic aperture was increased from 2.5 mm to 15.1 mm in steps of 0.70mm. FIG. 9B shows that the resolution improves and speckle sizedecreases according to theory. Specifically, speckle size and resolutionloss are inversely proportional to the size of the imaging aperture.Recovery was performed without the use of the denoising regularizer sothat an accurate measurement of the speckle size was obtained. Specklesize is measured as the full width half maximum of autocorrelationfunction of the intensity pattern (43). A region of the registrationsquare at the top of the target (the square between group 1, element 1and group 0, element 2) is chosen to measure the speckle size and thereported value is the average of both the vertical and horizontalprofiles. It should also be noted that the discrete nature of the USAFtarget causes the measured resolution to deviate from the predictedvalues. The measured speckle size decreases according to theoreticalpredictions, and demonstrates a 6× improvement in spatial resolution. InFIG. 9B, the measured and predicted values are shown in dashed and solidlines, respectively. Speckles size computed without use of the denoisingregularizer. The slight deviation of measured improvement is aconsequence of discretization of the resolution chart.

Section III Discussion: Considerations for Real-World Implementations

One important area of further study is modeling and accounting forobjects with low contrast. FIG. 10 illustrates the simulation ofsuper-resolving a complex object whose amplitude has varying brightnessratios. The owl is the brightest element (amplitude of 1), thebackground is the darkest element, and brightness of the text and camerais the average of the owl and background. The minimum amplitude isvaried from 0.1 to 0.5 and a simulated FP data capture is generatedmatching the experimental parameters used for the USAF and fingerprintdatasets (focal length=75 mm, aperture diameter=2.5 mm, overlap=72%,synthetic aperture size=15.1 mm).

Objects with strong contrast between the foreground and backgroundamplitude values have a higher fidelity reconstruction than objectswhere the background and foreground amplitudes are similar. FIG. 10provides the high-resolution complex object used for testing in the toprow. The bottom three rows represent reconstructions where the amplitudein the high resolution object is varied to different intensity ranges.As the contrast decreases in the amplitude of the high-resolution objectas shown in the left column of FIG. 10, a larger area of the capturedimages exhibits speckle. The reconstruction quality decreases as thecontrast increases as shown in the right column of FIG. 10. Speckleappears in the background and fine features that can be seen with arelatively high contrast (first row) are no longer resolvable (thirdrow).

When the amplitude of the background is non-zero, speckles will form.Removing the speckle from the background will require strongerregularization than the method presented in this paper, and is apromising avenue of research. Alternative strategies, such as destroyingtemporal coherence to reduce speckle contrast have been employed in FPmicroscopy (35), and may be of some benefit in near- to mid-rangemacroscopic imaging.

It should be noted that various changes and modifications to theembodiments described herein will be apparent to those skilled in theart. Such changes and modifications may be made without departing fromthe spirit and scope of the present invention and without diminishingits attendant advantages. For example, various embodiments of thesystems and methods may be provided based on various combinations of thefeatures and functions from the subject matter provided herein.

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We claim:
 1. A method for imaging objects comprising the steps of:providing an imaging device including a camera sensor, a camera lens,and a pupil; illuminating an object with the light source; receiving anillumination field reflected by the object, wherein an aperture fieldthat intercepts the pupil of the imaging system is an opticalpropagation of the illumination field at an aperture plane; receiving aportion of the aperture field onto a camera sensor; receiving a sensorfield of optical intensity on the image sensor, wherein the sensor fieldis a Fourier transform of the aperture field immediately after thecamera lens; iteratively centering the camera focus along the Fourierplane at different locations to produce a series of sensor fields;stitching together the sensor fields in the Fourier domain to generatean image; determining a plurality of phase information for each sensorfield in the series of sensor fields; and applying the plurality ofphase information to the image; receiving a plurality of illuminationfields reflected by the object; and denoising the intensity of pluralityof illumination fields using Fourier ptychography.
 2. The method ofclaim 1, wherein the optical propagation is one of a Fourier transform,a Fresnel transform, an angular spectrum propagation, and a Huygensconvolution.
 3. The method of claim 1, wherein the object includes anoptically rough surface such that the illumination fields of theplurality of illumination fields are out of phase, creating speckle. 4.The method of claim 3, further comprising the step of utilizing Fourierptychography to reduce speckle by increasing the aperture diameter. 5.The method of claim 3, further comprising the step of inserting arotating diffuser in an optical path before the object to reducespeckle.
 6. The method of claim 1, further comprising the step ofgenerating a grid of images to yield a synthetic aperture.
 7. The methodof claim 6, wherein a synthetic aperture of at least 14.5 mm isgenerated.
 8. The method of claim 1, including the steps of recordingimages with three different shutter times and joining the recordedimages together to yield a high-dynamic range image.
 9. The method ofclaim 1, wherein the step of determining a plurality of phaseinformation for each sensor field comprises iteratively estimating theimage intensity in the Fourier domain.